Exact Explicit Peakon and Periodic Cusp Wave Solutions for Several Nonlinear Wave Equations

We present the mixed dn-sn method for finding periodic wave solutions of some nonlinear wave equations. Introducing an appropriate transformation, we extend this method to a special type of nonlinear equations and construct their solutions, which are not expressible as polynomials in the Jacobi elliptic functions. The obtained solutions include the well known kink-type and bell-type solutions as a limiting cases. Also, some new travelling wave solutions are found. - PACS: 02.30.Jr; 03.40.Kf

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EXACT SOLUTIONS AND THEIR DYNAMICS OF TRAVELING WAVES IN THREE TYPICAL NONLINEAR WAVE EQUATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409024049 ◽

2009 ◽

Vol 19 (07) ◽

pp. 2249-2266 ◽

Cited By ~ 12

Author(s):

JIBIN LI ◽

YI ZHANG ◽

GUANRONG CHEN

Keyword(s):

Nonlinear Wave ◽

Traveling Waves ◽

Traveling Wave ◽

Wave Equations ◽

Visual Illusion ◽

Soliton Solutions ◽

Traveling Wave Solutions ◽

Dynamical Analysis ◽

Nonlinear Wave Equations ◽

Wave Solutions

It was reported in the literature that some nonlinear wave equations have the so-called loop- and inverted-loop-soliton solutions, as well as the so-called loop-periodic solutions. Are these true mathematical solutions or just numerical artifacts? To answer the question, this article investigates all traveling wave solutions in the parameter space for three typical nonlinear wave equations from a theoretical viewpoint of dynamical systems. Dynamical analysis shows that all these loop- and inverted-loop-solutions are merely visual illusion of numerical artifacts. To reveal the nature of such special phenomena, this article also offers the mathematical parametric representations of these traveling wave solutions precisely in analytic forms.

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Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method

Zeitschrift für Naturforschung A ◽

10.1515/zna-2015-0122 ◽

2015 ◽

Vol 70 (11) ◽

pp. 969-974 ◽

Cited By ~ 10

Author(s):

Melike Kaplan ◽

Arzu Akbulut ◽

Ahmet Bekir

Keyword(s):

Nonlinear Wave ◽

Wave Equations ◽

Travelling Wave ◽

Nonlinear Evolution Equations ◽

Equation Method ◽

Nonlinear Wave Equations ◽

Auxiliary Equation ◽

Travelling Wave Solutions ◽

Auxiliary Equation Method ◽

Wave Solutions

AbstractThe auxiliary equation method presents wide applicability to handling nonlinear wave equations. In this article, we establish new exact travelling wave solutions of the nonlinear Zoomeron equation, coupled Higgs equation, and equal width wave equation. The travelling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Throughout the article, all calculations are made with the aid of the Maple packet program.

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